Numerical modeling of internal tides and submesoscale turbulence in the US Caribbean regional ocean

The US Caribbean ocean circulation is governed by an influx of Atlantic water through the passages between Puerto Rico, Hispaniola and the Virgin Islands, and an interplay of the Caribbean Sea water with the local topography of the region. We present an analysis of the US Caribbean ocean flow simulated by the USCROMS; which is the ROMS AGRIF model configured for the US Caribbean regional ocean at a horizontal resolution of 2 km. Outputs from the USCROMS show a seasonal variability in the strength of submesoscale turbulence within a mixed layer whose depth varies from −70 to −20 m from winter to summer, and internal tides originating from the passages between the islands. Energy spectra of the simulated baroclinic velocity show diurnal and semi-diurnal maxima and several higher-order harmonic frequency maxima associated with non-linear internal waves forming over steep slopes with super-critical topography in the continental shelf. The strongest conversion rates of the depth-averaged barotropic to baroclinic tidal energy occur at localized regions in the continental shelf with super-critical topography. These regions also exhibit enhanced transport and dissipation of the depth-averaged barotropic and baroclinic tidal kinetic energy. The dissipation in these regions is nearly 3 orders of magnitude stronger than the open ocean dissipation. The energy transport terms show a seasonal pattern characterized by stronger variance during summer and reduced variance during the winter. At the benthic regions, the dissipation levels depend on the topographic depth and the tidal steepness parameter. If the benthic region lies within the upper-ocean mixed-layer, the benthic dissipation is enhanced by surface-forced processes like wind forcing, convective mixing, submesoscale turbulence and bottom friction. If the benthic region lies below the mixed-layer, the benthic dissipation is enhanced by the friction between the super-critical topographic slopes and the periodically oscillating baroclinic tidal currents. Due to bottom friction, the tidal oscillation in the lateral currents adjacent to the sloping topography generates cyclonic and anti-cyclonic vortices with O(1) Rossby number depending on the orientation of the flow. While the cyclonic vortices form positive potential vorticity (q) leading to barotropic shear instability, anti-cyclonic vortices form negative q which leads to periodically occurring inertial instability. The lateral and inertial instabilities caused by the baroclinic tidal oscillations act as routes to submesoscale turbulence at the benthic depths of −100 m to −400 m near the super-critical topography of the continental shelf, forming O(10 km) long streaks of turbulent water with dissipation levels that are 3 orders of magnitude stronger than the dissipation in the open ocean at the same depth. The magnitudes of the dissipation and q at the benthic regions over super-critical continental-shelf topography are also modulated by the spring-neap tidal signals.

) for the high-pass filtered conversion term C hp , and the pressure transport terms P hp SSH , P hp t , P hp c respectively. These variances are calculated instantaneously, and then averaged over each of the 12 months. The monthly averaging was done to show the seasonal variability prominently.

Appendices Appendix A Normalized topographic gradient
We calculate the normalized topographic gradient as where h denotes the topographic depth and h max is the depth of the deepest region in the domain. Thus, δh/h is inverse proportional to the topographic depth and directly proportional to the horizontal gradient. We use this quantity as a metric to determine the regions with strong magnitudes of tidal kinetic energy production.

Appendix B Tidal parameters
The steepness parameter ϕ is defined as the ratio of the topographic gradient to the slope of the internal wave phase where h is the topographic depth, and k/m = ω 2 −f 2 is the ratio of the horizontal to the vertical wavenumber for an internal wave with frequency ω propagating in stratified water with buoyancy frequency N 2 . The steepness parameter ϕ is used to distinguish between sub-critical (ϕ < 1), critical (ϕ = 1), and super-critical (ϕ > 1) topography.
Tidal excursion parameter is defined as the ratio of the length-scale covered by the barotropic tidal velocity amplitude, to the topographic length-scale. Mathematically, the excursion parameter ψ is expressed as where U 0 is the amplitude of the barotropic tidal velocity, ω is the tidal frequency and λ is the topographic length scale which is a representative of the horizontal dimension of the bottom mount. Following [1] and [2], we estimate the topographic length-scale as λ = |h max − h|/ (∂h/∂x) 2 + (∂h/∂y) 2 where h max is the depth of the deepest region in the domain. Thus, |h max −h| gives an estimate of the vertical dimension of the topographic mount from the deepest point.

Appendix C Barotropic and Baroclinic kinetic energy
The Navier-Stokes [3] prognostic equation for the velocity is expressed in tensor notation as where u i is the velocity along the direction denoted by i, and x j is the spatial scale along the direction j. The term 2ϵ ijk σ j u k denotes the Coriolis force. On the right hand side, the variable η denotes the sea-surface height, p ′ is the baroclinic pressure perturbation, ρ ′ is the density perturbation, and ν ij is the eddy viscosity for the velocity u i along direction j. The perturbation pressure is expressed as where p and p 0 are the total pressure and the ambient equilibrium pressure respectively. The perturbation density is expressed as The diffusion of velocity is expressed as ∂ ∂xj ν ij ∂ui ∂xj . At the topmost face, the vertical eddy viscosity ν i3 and surface wind stress τ 1,2 (unit N/m 2 ) are related by the following boundary condition At the bottom face, quadratic bottom drag replaces the diffusion term in the bottom boundary condition, expressed as where c d = is the bottom drag coefficient, κ is the Von-Karman constant, and z 0 is the bottom roughness length parameter.
The current velocity u i can be expressed as u i = U i +ȗ i , where U i and u i are the barotropic and baroclinic components respectively. The barotropic component U i=1,2 is constant throughout the depth of the water column.
To obtain the baroclinic kinetic energy, we begin by depth-averaging the Navier-Stokes equation from the topographic depth to the surface. The depthaveraging is done in the following manner: where Λ is the variable at the center of each grid cell, δz k is the vertical grid cell thickness, N is the number of grid cells along the vertical (in our case N = 32), and Σ denotes the summation of all vertical grid cells from 1 to N . The overline denotes the depth-average. Multiplying equation C4 with the total velocity u i and averaging over the entire depth, we get a prognostic equation for the depth-averaged total kinetic energy 1 2 u 2 i , given as where, the overline denotes the depth average. Now, we multiply equation C4 with the barotropic velocity component U i=1,2 to get a prognostic equation for the barotropic kinetic energy 1 2 U 2 i , where i = 1, 2 denotes only the horizontal components. The barotropic kinetic energy (E BT K ) prognostic equation using the tensor i = 1, 2 is given as where, ADV t is the depth-averaged advective work of E BT K , and St is the depth-averaged product of the baroclinic stress divergence and the barotropic velocity. The terms Dt and ϵ t are the depth-averaged diffusion and dissipation of E BT K respectively. The depth-averaged diffusion term Dt is given as and the dissipation ϵ t is expressed as Subtracting the E BT K prognostic equation C9 from the total kinetic energy equation C8, we obtain a depth-averaged prognostic equation for the baroclinic kinetic energy E BCK , expressed as 1 2 (ȗ 2 1,2 + u 2 3 ) Thus, the equation for E BCK is given as ∂ ∂t where the terms Dc and ϵ c denote the depth-averaged diffusion of E BCK and the dissipation of E BCK respectively. The baroclinic stress termȗ 1,2ȗ2,1 is analogous to the turbulent Reynolds stress obtained by Reynolds averaging of the Navier-Stokes equation. The term Sc represents the extraction of energy from the barotropic lateral shear by the baroclinic lateral stress. Since the depth averaging is done throughout the entire water column, the eddy viscosities at the surface and bottom faces are replaced by the corresponding boundary conditions for the wind stress (equation C5) and bottom stress (equation C6) respectively. The depth-averaged diffusion of E BCK is Dc, and is given as The depth-averaged dissipation of E BCK is ϵ c , and is given as The ϵ c is a summation of the horizontal component ϵ h parameterized with lateral shear (j = 1, 2), and the vertical component ϵ v parameterized using vertical shear (j = 3). and where the tensors i, j represent only the horizontal directions. All flux quantities are depth-averaged. For the baroclinic kinetic energy E BCK , the horizontal pressure energy flux (F P c ) and the advective energy flux (F ADV c ) are given as and

Appendix E Conversion of the net barotropic to baroclinic energy
The net barotropic energy is the summation of the barotropic kinetic energy E BT K and the perturbation potential energy due to perturbations in the seasurface height. The net baroclinic energy is the summation of the baroclinic kinetic energy E BCK and the available potential energy due to perturbations in the interior density surfaces. The rate of conversion from the net barotropic to net baroclinic energy, when averaged over the depth of the water column, is given as C = 1 ρ0 ρ ′ gW [4,5], where W is the barotropic convergence rate, obtained by vertically integrating the equation of continuity.
where U i is the barotropic horizontal velocity, h is the bottom depth, and η is the sea-surface height [2,4].

Appendix F Potential Vorticity
In a hydrostatic numerical setup, the Ertel's potential vorticity q is expressed as